The Stacks project

Lemma 31.18.1. Let $f : X \to S$ be a morphism of schemes. Let $D \subset X$ be a closed subscheme. Assume

  1. $D$ is an effective Cartier divisor, and

  2. $D \to S$ is a flat morphism.

Then for every morphism of schemes $g : S' \to S$ the pullback $(g')^{-1}D$ is an effective Cartier divisor on $X' = S' \times _ S X$ where $g' : X' \to X$ is the projection.

Proof. Using Lemma 31.13.2 we translate this as follows into algebra. Let $A \to B$ be a ring map and $h \in B$. Assume $h$ is a nonzerodivisor and that $B/hB$ is flat over $A$. Then

\[ 0 \to B \xrightarrow {h} B \to B/hB \to 0 \]

is a short exact sequence of $A$-modules with $B/hB$ flat over $A$. By Algebra, Lemma 10.39.12 this sequence remains exact on tensoring over $A$ with any module, in particular with any $A$-algebra $A'$. $\square$

Comments (2)

Comment #1456 by jojo on

In the statement should be (or the other way around).

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View Lemma 31.18.1 as pdf